Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems

Previous Article
Global controllability and stabilizability of Kawahara equation on a periodic domain
MCRF Home
This Issue
Next Article

June 2015, 5(2): 359-376. doi: 10.3934/mcrf.2015.5.359

Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems

Guangliang Zhao ^1, , Fuke Wu ^2, and George Yin ^3,

1.	GE Global Research, 1 Research Circle, Niskayuna, NY 12309, United States
2.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074
3.	Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Received January 2014 Revised February 2014 Published April 2015

Abstract
Related Papers

To treat networked systems involving uncertainty due to randomness with both continuous dynamics and discrete events, this paper focuses on diffusions modulated by a continuous-time Markov chain. In our paper [19], we considered ordinary differential equations with Markovian switching. This paper further treats more complex cases, namely, stochastic differential equations with Markovian switching. Our goal is to stabilize the systems under consideration. One of the difficulties is that the systems grow much faster than the allowable rates in the literature of stochastic differential equations. As a result, the underlying systems have finite explosion time. To overcome the difficulties, we develop feedback controls to extend the local solutions to global solutions and to stabilize the resulting systems. The feedback controls are Brownian type of perturbations. We establish the existence of global solution, prove the stability of the resulting systems, obtain boundedness in probability as $t\to\infty$, and provide sufficient conditions for almost sure stability. Then we present numerical examples to illustrate the main results.

Keywords: stabilization, feedback control., regularity, stability, Regime-switching diffusion.

Mathematics Subject Classification: Primary: 60J60, 60J27, 93E15; Secondary: 93E0.

Citation: Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control & Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359

References:

[1]	J. A. D. Appleby and X. Mao, Stochastic stabilisation of functional differential equations,, Systems & Control Letters, 54 (2005), 1069. doi: 10.1016/j.sysconle.2005.03.003. Google Scholar
[2]	J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise,, IEEE Trans. Automat. Control, 53 (2008), 683. doi: 10.1109/TAC.2008.919255. Google Scholar
[3]	L. Arnold, H. Crauel and V. Wihstusz, Stabilization of linear system by noise,, SIAM J. Control Optim., 21 (1983), 451. doi: 10.1137/0321027. Google Scholar
[4]	A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364. doi: 10.1016/j.jmaa.2003.12.004. Google Scholar
[5]	F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth,, Systems & Control Letters, 57 (2008), 262. doi: 10.1016/j.sysconle.2007.09.002. Google Scholar
[6]	M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1952. doi: 10.1137/S0363012996299302. Google Scholar
[7]	R. Z. Khasminskii, Stochastic Stability of Differential Equations,, $2^{nd}$ edition, (2012). doi: 10.1007/978-3-642-23280-0. Google Scholar
[8]	R. Z. Khasminskii and G. Yin, Asymptotic behavior of parabolic equations arising from null-recurrent diffusions,, J. Differential Eqs., 161* (2000), 154. doi: 10.1006/jdeq.1999.3647. Google Scholar*
[9]	H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, $2^{nd}$ edition, (2003). doi: 10.1007/b97441. Google Scholar
[10]	B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models,, J. Math. Anal. Appl., 339 (2008), 419. doi: 10.1016/j.jmaa.2007.06.058. Google Scholar
[11]	R. Liptser and A. N. Shiryaev, Theory of Martingale,, Kluwer Academic Publishers, (1989). doi: 10.1007/978-94-009-2438-3. Google Scholar
[12]	X. Mao, Stability of stochastic differential equations with Markovian switching,, Stochastic Process. Appl., 79 (1999), 45. doi: 10.1016/S0304-4149(98)00070-2. Google Scholar
[13]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006). doi: 10.1142/9781860948848_fmatter. Google Scholar
[14]	X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008). doi: 10.1533/9780857099402. Google Scholar
[15]	A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations,, Amer. Math. Soc., (1989). Google Scholar
[16]	F. Wu and S. Hu, Suppression and stabilisation of noise,, Internat. J. Control, 82 (2009), 2150. doi: 10.1080/00207170902968108. Google Scholar
[17]	G. Yin, X. R. Mao, C. Yuan and D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: Numerical algorithms and existence and uniqueness of solutions,, SIAM J. Math. Anal., 41 (2010), 2335. doi: 10.1137/080727191. Google Scholar
[18]	G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010). doi: 10.1007/978-1-4419-1105-6. Google Scholar
[19]	G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems,, SIAM J. Appl. Math., 72 (2012), 1361. doi: 10.1137/110851171. Google Scholar
[20]	C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154. doi: 10.1016/j.jmaa.2009.03.066. Google Scholar

show all references

References:

[1]	J. A. D. Appleby and X. Mao, Stochastic stabilisation of functional differential equations,, Systems & Control Letters, 54 (2005), 1069. doi: 10.1016/j.sysconle.2005.03.003. Google Scholar
[2]	J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise,, IEEE Trans. Automat. Control, 53 (2008), 683. doi: 10.1109/TAC.2008.919255. Google Scholar
[3]	L. Arnold, H. Crauel and V. Wihstusz, Stabilization of linear system by noise,, SIAM J. Control Optim., 21 (1983), 451. doi: 10.1137/0321027. Google Scholar
[4]	A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364. doi: 10.1016/j.jmaa.2003.12.004. Google Scholar
[5]	F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth,, Systems & Control Letters, 57 (2008), 262. doi: 10.1016/j.sysconle.2007.09.002. Google Scholar
[6]	M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1952. doi: 10.1137/S0363012996299302. Google Scholar
[7]	R. Z. Khasminskii, Stochastic Stability of Differential Equations,, $2^{nd}$ edition, (2012). doi: 10.1007/978-3-642-23280-0. Google Scholar
[8]	R. Z. Khasminskii and G. Yin, Asymptotic behavior of parabolic equations arising from null-recurrent diffusions,, J. Differential Eqs., 161* (2000), 154. doi: 10.1006/jdeq.1999.3647. Google Scholar*
[9]	H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, $2^{nd}$ edition, (2003). doi: 10.1007/b97441. Google Scholar
[10]	B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models,, J. Math. Anal. Appl., 339 (2008), 419. doi: 10.1016/j.jmaa.2007.06.058. Google Scholar
[11]	R. Liptser and A. N. Shiryaev, Theory of Martingale,, Kluwer Academic Publishers, (1989). doi: 10.1007/978-94-009-2438-3. Google Scholar
[12]	X. Mao, Stability of stochastic differential equations with Markovian switching,, Stochastic Process. Appl., 79 (1999), 45. doi: 10.1016/S0304-4149(98)00070-2. Google Scholar
[13]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006). doi: 10.1142/9781860948848_fmatter. Google Scholar
[14]	X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008). doi: 10.1533/9780857099402. Google Scholar
[15]	A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations,, Amer. Math. Soc., (1989). Google Scholar
[16]	F. Wu and S. Hu, Suppression and stabilisation of noise,, Internat. J. Control, 82 (2009), 2150. doi: 10.1080/00207170902968108. Google Scholar
[17]	G. Yin, X. R. Mao, C. Yuan and D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: Numerical algorithms and existence and uniqueness of solutions,, SIAM J. Math. Anal., 41 (2010), 2335. doi: 10.1137/080727191. Google Scholar
[18]	G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010). doi: 10.1007/978-1-4419-1105-6. Google Scholar
[19]	G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems,, SIAM J. Appl. Math., 72 (2012), 1361. doi: 10.1137/110851171. Google Scholar
[20]	C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154. doi: 10.1016/j.jmaa.2009.03.066. Google Scholar

[1]	Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control & Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022
[2]	Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048
[3]	Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control & Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237
[4]	Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control & Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21
[5]	Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2019072
[6]	Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100
[7]	Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019036
[8]	Martin Gugat, Mario Sigalotti. Stars of vibrating strings: Switching boundary feedback stabilization. Networks & Heterogeneous Media, 2010, 5 (2) : 299-314. doi: 10.3934/nhm.2010.5.299
[9]	Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079
[10]	Jiapeng Liu, Ruihua Liu, Dan Ren. Investment and consumption in regime-switching models with proportional transaction costs and log utility. Mathematical Control & Related Fields, 2017, 7 (3) : 465-491. doi: 10.3934/mcrf.2017017
[11]	Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2018166
[12]	Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132
[13]	Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063
[14]	Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085
[15]	Thomas I. Seidman. Optimal control of a diffusion/reaction/switching system. Evolution Equations & Control Theory, 2013, 2 (4) : 723-731. doi: 10.3934/eect.2013.2.723
[16]	Cătălin-George Lefter, Elena-Alexandra Melnig. Feedback stabilization with one simultaneous control for systems of parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 777-787. doi: 10.3934/mcrf.2018034
[17]	Caojin Zhang, George Yin, Qing Zhang, Le Yi Wang. Pollution control for switching diffusion models: Approximation methods and numerical results. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3667-3687. doi: 10.3934/dcdsb.2018310
[18]	Thomas I. Seidman, Houshi Li. A note on stabilization with saturating feedback. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 319-328. doi: 10.3934/dcds.2001.7.319
[19]	A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289
[20]	Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303

2018 Impact Factor: 1.292

American Institute of Mathematical Sciences